Finite semigroups with few term operations

Abstract

For a semigroup S, pn(S) denotes the number of n-ary term operations of S depending on all their variables. The purpose of this paper is to study finite semigroups S with the property that their pn-sequence p(S)=〈p0(S), p1(S),...〉 is bounded. Such semigroups are described first in terms of identities and then structurally as nilpotent extensions of semilattices, Boolean groups and rectangular bands. As a corollary it is shown that if p(S) is bounded then eventually either pn(S)=0 or 1. It is also shown that there is an effective procedure which decides whether the pn-sequence of a given finite semigroup is bounded or not. © 2001 Elsevier Science B.V.